On 21 Jul 2009, at 07:22, Rex Allen wrote:

>
> Brent, I intend to reply more directly to your post soon, as I think
> there's a lot to be said in response.

I agree! I let you comment first.


>
>
> But in the meantime:
>
> So I just finished reading David Deutsch's "The Fabric of Reality",
> and I'm curious what you (Brent, Bruno, and anyone else) make of the
> following passage at the end of chapter 10, The Nature of Mathematics.
> The first paragraph is at least partly applicable to Brent's recent
> post, and the second seems relevant to Bruno's last response.  It
> makes one wonder what other darkly esoteric abstractions may stalk the
> abyssal depths of Platonia???
>
> The passage:
>
> "Mathematical entities are part of the fabric of reality because they
> are complex and autonomous.  The sort of reality they form is in some
> ways like the realm of abstractions envisaged by Plato or Penrose:
> although they are by definition intangible, they exist objectively and
> have properties that are independent of the laws of physics.

OK. Note that assuming comp, the laws of physics are dependent of the  
math.




> However,
> it is physics that allows us to gain knowledge of this realm.

This is a physicalist assumption.




> And it
> imposes stringent constraints.

Assuming comp, those constraints are themselves a mathematical origin.




> Whereas everything in the physical
> reality is comprehensible,

Everything? This is an assumption (and is probably wrong in the comp  
frame).



> the comprehensible mathematical truths are
> precisely the infinitesimal minority which happen to correspond
> exactly to some physical truth - like the fact that if certain symbols
> made of ink on paper are manipulated in certain ways, certain other
> symbols appear.  That is, they are the truths that can be rendered in
> virtual reality.

This follows from comp.



> We have no choice but to assume that the
> incomprehensible mathematical entities are real too, because they
> appear inextricably in our explanations of the comprehensible ones.

They appear in the mind or dreams of the universal machine. Here the  
comp hyp. makes possible to distinguish ontological mathematics (no  
need to take more than a tiny part of arithmetic), and the  
epistemological mathematics, which has no mathematically definable  
bound.



>
>
> There are physical objects - such as fingers, computers and brains -
> whose behaviour can model that of certain abstract objects.  In this
> way the fabric of physical reality provides us with a window on the
> world of abstractions.

Physicalist assumption. With comp the physical world emerges itself  
from a statistical sum on infinitely many computations.



> It is a very narrow window and gives us only a
> limited range of perspectives.  Some of the structures that we see out
> there, such as the natural numbers or the rules of inference of
> classical logic, seem to be important or 'fundamental' to the abstract
> world, in the same way as deep laws of nature are fundamental to the
> physical world.

Yes. Comp explains this, and exploits this.



> But that could be a misleading appearance.  For what
> we are really seeing is only that some abstract structures are
> fundamental to our understanding of abstractions.  We have no reason
> to suppose that those structures are objectively significant in the
> abstract world.

Comp does make them significant.


>  It is merely that some abstract entities are nearer
> and more easily visible from our window than others."

Comp explains this.

I appreciate very much the FOR book, but Deutsch does not take into  
account the fact that if we are digitalizable machines, our  
predictions have to rely eventually on the infinitely many relations  
between numbers. From the first person point of view, those relations  
rely themselves on many infinities which goes beyond elementary  
arithmetic.

With the comp assumption, we have a simple theory of everything:  
elementary arithmetic (without the induction axioms). In that theory  
we can prove the existence of universal machine, and their (finite)  
pieces of dreams, and why those machines will, from their own point of  
view infer the "induction axioms" and glue their dreams in projecting  
physical universe. Comp makes a tiny part of arithmetic a virtual  
"matrix" or "video game", which viewed from inside, will seem as a  
locally concrete reality. Problem: there could be too much "white  
rabbits", and other non computable manifestations predictable in our  
neighborhood. It could be no more than the 'quantum indeterminacy',  
but this remain to be completely proved (a part of this has been  
verified though).

Note that the epistemology is far richer than the ontology. The 'first  
person plenitude' (cf George Levy) is MUCH bigger than the minimal  
third person reality we need to explain the origin of the appearances.

Bruno

http://iridia.ulb.ac.be/~marchal/




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